Automatically Scale and Export All Files in Directory?
Using only Python and Blender
For clarity and security, I don't overwrite the original files.
This script will check for the scene files (which contain the object). Then, for each file separately, it will append all objects, scale them and save itself as the scaled file version. After that export the .fbx and finally on completion save itself as the original conversion script file.
With the script saved in the file convert.blend. My folder structure looks like this at the start.

Opening the file convert.blend and pressing Run Script the scaled scene files and fbx files will be created.
import bpy, os
import bmesh
import math
import mathutils

def clear_objects():
    for obj in bpy.data.objects:
        bpy.context.scene.objects.unlink(obj)
        obj.user_clear()
        bpy.data.objects.remove(obj)
    for me in bpy.data.meshes:
        me.user_clear()
        bpy.data.meshes.remove(me)

def save_scaled_objects(dir_source, dir_target, dir_fbx, scale = 1):
    dir_root = bpy.path.abspath(&quot;//&quot;)
    path_origin = bpy.data.filepath

    dir_source = os.path.join(dir_root, dir_source)
    dir_target = os.path.join(dir_root, dir_target)
    dir_fbx = os.path.join(dir_root, dir_fbx)

    files = os.listdir(dir_source)

    scn = bpy.context.scene

    for f in files:
        input_filepath = os.path.join(dir_source, f)
        output_filepath = os.path.join(dir_target, f)
        output_fbx = os.path.join(dir_fbx, os.path.splitext(f)[0] + &quot;.fbx&quot;)

        print(&quot;DIR&quot;,output_fbx)
        # append object data block
        with bpy.data.libraries.load(input_filepath, link=False) as (data_from, data_to):
            data_to.objects = data_from.objects
        #link all objects to current scene
        for obj in data_to.objects:
            if obj is not None:
                scn.objects.link(obj)
                #scale the objects data
                scale = 0.1
                for v in obj.data.vertices:
                    v.co *= scale
        print(output_filepath)
        # saves the file
        bpy.ops.wm.save_mainfile(filepath=output_filepath)
        #export to fbx
        bpy.ops.export_scene.fbx(filepath = output_fbx)
        
        clear_objects()
    # on completion of all saves, restore the original path of this file
    bpy.ops.wm.save_mainfile(filepath=path_origin)


dir_source = &quot;scn_files&quot;
dir_target = &quot;scn_files_scaled&quot;
dir_fbx = &quot;fbx_export&quot;
save_scaled_objects(dir_source, dir_target, dir_fbx, scale = 0.1)


Linking and Appending
Remove Objects
Base Path, Path Joining, Splitext